In $R^n$, the Laplace-Beltrami operator is just the Laplacian, and its eigenstructure is well known. There are also explicit expressions for the eigenvalues/eigenvectors of the Laplace-Beltrami operator on the sphere.

Question: Are there any other nontrivial surfaces for which explicit expressions
for the eigenvalues/eigenvectors of the Laplace-Beltrami operator have
been worked out ? I was unable to even find anything for an ellipsoid.

I also wanted to emphasize that I'm looking for closed form expressions.

My guess is that in general good things happen if your manifold is a homogeneous space for a Lie group (with metric induced from an invariant metric on the Lie group) and otherwise there doesn't seem to be enough structure to say much. I suppose hyperbolic surfaces are almost homogeneous spaces, but the eigenvalues of the Laplacian on some hyperbolic surfaces are actually related to some deep number theory (see en.wikipedia.org/wiki/Selberg's_conjecture).
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Qiaochu YuanJun 7 '12 at 22:57

2 Answers
2

The answer is nice for flat tori in every dimension. Let's write such a torus as $V/\Gamma$ where $V$ is a finite-dimensional real vector space of dimension $n$ with inner product $\langle \cdot, \cdot \rangle$ and $\Gamma$ is a lattice (a discrete subgroup isomorphic to $\mathbb{Z}^n$ which spans $V$). Any twice-differentiable eigenfunction $f : V/\Gamma \to \mathbb{C}$ of the Laplacian is in particular a bounded eigenfunction of the Laplacian on $V$, so we can take it to have the form
$$f_w(v) = e^{2 \pi i \langle w, v \rangle}$$

for some $w \in V$ (for reasons to be described later). We also need to impose the constraint that $f_w$ is invariant under $\Gamma$, hence that
$$e^{2\pi i \langle w, v \rangle} = e^{2 \pi i \langle w, v + g \rangle}$$

for every $g \in \Gamma$. This condition is satisfied if and only if $w$ belongs to the dual lattice $\Gamma^{\vee}$, which consists of all vectors $w$ such that $\langle w, g \rangle \in \mathbb{Z}$ for all $g \in \Gamma$. Moreover,
$$\Delta f_w = - 4 \pi^2 \| w \|^2$$

so the eigenvalues of the Laplacian on $V/\Gamma$ are just $- 4\pi^2$ times the squares of the lengths of the vectors in $\Gamma^{\vee}$.

In general it's difficult to find the eigenvalues and the eigenfunctions of the Laplace-Beltrami operator of a general surface. However, something is known. In particular, the eigenvalues and eigenfunctions of the standard $n$-dimensional sphere are known. More precisely, the eigenfunctions are given by the spherical harmonics. If you search "laplacian eigenvalue sphere" on the Internet, you can find many references.

Also, if I remember correctly, eigenvalues of flat torus are known (references needed). Also, the eigenvalues of the Heisenberg manifold are known (see here).